Finite Group Databasebeta

← #10698331

Subgroups — #10698331

Order 768.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73398C211yesmaximal, characteristic
2#70792C212yesmaximal
3#69441C211yesmaximal, characteristic
4C3×#1274C211yesmaximal, characteristic
5#67746C212yesmaximal
6#10373—31nomaximal, Sylow
7#5470C2211yescharacteristic
8C3×#431C2212yes—
9C3×#376C2211yescharacteristic
10C4×C4⋊Dic3C2212yes—
11C4×Dic3⋊C4—22no—
12#4924C2211yescharacteristic
13#4660—22no—
14#4657—22no—
15#4523—22no—
16#4467—22no—
17#4467—22no—
18#2402—31no—
19#1995—31no—
20#1503—32no—
21#1303—32no—
22#1274D311yescharacteristic
23C2×C3×C42D412yes—
24C2×C3×C42C2311yescharacteristic
25#926—21no—
26C2×C4⋊Dic3—22no—
27#924—22no—
28C2×Dic3⋊C4—22no—
29C2×Dic3⋊C4—22no—
30C2×C4×Dic3—22no—
31C2×#208—21no—
32#870—21no—
33#869Q812yes—
34C4×Dic6—22no—
35C3×C4⋊C8—22no—
36C3×C22⋊C8—22no—
37C3×C4×C8—22no—
38C3×#95—22no—
39C3⋊#95—22no—
40C3⋊#95—22no—
41C3⋊#95D412yes—
42C3⋊#95D412yes—
43C4⋊(C3⋊C8)—22no—
44#533—31no—
45#431C2×D312yes—
46#423—31no—
47#400—62no—
48#398—62no—
49C4×C4⋊C4—32no—
50C4×C4⋊C4—62no—
51#376C2×D311yescharacteristic
52#339—62no—
53#328—62no—
54#328—62no—
55C22×C3×C4—22no—
56C22×C3×C4C2×D411yescharacteristic
57C22×C3×C4C2×D412yes—
58C22×Dic3—22no—
59C22×Dic3—22no—
60C2×Dic6—22no—
61C2×C3×C8—42no—
62C3×C42—22no—
63C3×C42—22no—
64C3×C42C2×Q811yescharacteristic
65C3×C42#4112yes—
66C3×C42#4111yescharacteristic
67#217—41no—
68C4⋊Dic3—22no—
69C4⋊Dic3—22no—
70C4⋊Dic3—44no—
71Dic3⋊C4—44no—
72Dic3⋊C4—44no—
73Dic3⋊C4—42no—
74C4×Dic3—41no—
75C4×Dic3—42no—
76C4×Dic3—42no—
77#208—42no—
78C2×C3⋊C8—41no—
79C2×M16—61no—
80C4⋊Q8—32no—
81#125—61no—
82C22⋊Q8—62no—
83C4×Q8—62no—
84#117—61no—
85C2×C4⋊C4—62no—
86C2×C4⋊C4—62no—
87C2×C4⋊C4—62no—
88C2×C42—62no—
89C2×C42D1212yes—
90C2×C42C22×D311yescharacteristic
91C4⋊C8—22no—
92C4⋊C8—62no—
93C22⋊C8—22no—
94C4×C8—22no—
95#95—22no—
96#95—32no—
97#95—32no—
98#95—62no—
99#95—62no—
100C23×C3#12011yescharacteristic
101C2×C3×C4—42no—
102C2×C3×C4—42no—
103C2×C3×C4—42no—
104C2×C3×C4—42no—
105C2×C3×C4C22⋊Q812yes—
106C2×C3×C4C22⋊Q811yescharacteristic
107C2×C3×C4#12411yescharacteristic
108C2×C3×C4#12412yes—
109C2×C3×C4#13512yes—
110C2×C3×C4#13712yes—
111C2×Dic3—24no—
112C2×Dic3—42no—
113C2×Dic3—42no—
114C2×Dic3—42no—
115C2×Dic3—42no—
116C2×Dic3—42no—
117C2×Dic3—42no—
118Dic6—82no—
119C3×C8—82no—
120C3⋊C8—81no—
121C2×Q8—62no—
122C22×C4—22no—
123C22×C4—62no—
124C22×C4—62no—
125C22×C4C2×D1211yescharacteristic
126C22×C4D3×D412yes—
127M16—122no—
128C2×C8—42no—
129C2×C8—121no—
130C4⋊C4—62no—
131C4⋊C4—62no—
132C4⋊C4—124no—
133C4⋊C4—124no—
134C4⋊C4—124no—
135C4⋊C4—122no—
136C22⋊C4—121no—
137C42—22no—
138C42—22no—
139C42—121no—
140C42—122no—
141C42—122no—
142C42#23512yes—
143C42#23711yescharacteristic
144C42D3×Q811yescharacteristic
145C22×C3—22no—
146C22×C3#39311yescharacteristic
147C22×C3#45212yes—
148C3×C4—22no—
149C3×C4—22no—
150C3×C4—82no—
151C3×C4#47912yes—
152C3×C4#48811yescharacteristic
153C3×C4#49111yescharacteristic
154Dic3—44no—
155Dic3—82no—
156Dic3—82no—
157C23#88211yescharacteristic
158Q8—242no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—64no—
164C2×C4—122no—
165C2×C4—122no—
166C2×C4—122no—
167C2×C4—122no—
168C2×C4—122no—
169C2×C4—122no—
170C2×C4#87511yescharacteristic
171C2×C4#88512yes—
172C2×C4#89612yes—
173C2×C4#89711yescharacteristic
174C2×C4#90412yes—
175C2×C4#90912yes—
176C8—82no—
177C8—241no—
178C2×C3—22no—
179C2×C3#202611yescharacteristic
180C2×C3#203112yes—
181C22—22no—
182C22#467311yescharacteristic
183C22#473312yes—
184C4—22no—
185C4—22no—
186C4—82no—
187C4—124no—
188C4—242no—
189C4—242no—
190C4#469711yescharacteristic
191C4#470311yescharacteristic
192C4#482912yes—
193C3#1037311yescharacteristic, complemented, Sylow
194C2—22no—
195C2#6869011yescharacteristic
196C2#6871412yes—
subgroupG/H
centerC2×C4#875
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1274C2
socleC22×C3#393
prime radicalC23×C3#120
hypercenter#1274D3
hypocenterC3#10373
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
193. C3#10373semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#733982
#707922
#694412
C3×#12742
#677462
#103733